You all are way too pessimistic.
The people who fail aren't the people who mess up. The people who fail are the ones who don't go head-first into what they want to do.
You get confused, because all these great mathematicians seem to have perfect records. You think that their greatness caused them to have perfect records. The thing is, the causation is the other way around. They had perfect records because they figured out EARLY to go head first. Then they were rewarded along the way with confirmation and support because of their early achievements, which led them to go even harder into it (thus making their record even more impressive).
When someone starts off on the right foot, they have every advantage.
The inverse is also true. If someone starts off on the wrong foot, they have every disadvantage. First and foremost, they don't have the support of anyone. This leads them to be hesitant, and not devote so much time into it, because failure seems likely. It's a self-fulfilling prophecy.
Stephen Smale is a perfect example of someone who was given a chance, and reversed his momentum. If you guys think it's just a blip in the radar, that's fine, but I think that's a terrible way to look at it.
It's completely ridiculous to think of mistakes in the past as damning to the future. Everyone screws up now and then. Some people screw up majorly. But you know what the worst mistake you could ever make is? To be afraid of screwing up.
There are too many math majors at my university who only take the basic courses, and let other people decide what they are going to learn. It's way too common. And you know what? These are the people who are really failing. Even if they get good grades, and graduate with a 4.0, they're failing badly - very badly.
So STOP IT!

Stop promoting the idea that you need to be brilliant or perfect or whatever to do something great. It's the wrong way to think about things.